A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces

Archive for Rational Mechanics and Analysis

Volumn 174, Issue 1, 2004, Pages 49-81

  Quittner, P ^a   Souplet, Ph ^b  

^a Physics and Informatics   (Slovakia)

^b CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 7244227690     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00205-004-0323-8     Document Type: Article

References (37)

1. AMBROSIO, L., DANCER, N.: Calculus of variations and partial differential equations. Springer, Berlin 1999
2. ALVES, C.O., DE FIGUEIREDO, D.G.: Nonvariational elliptic systems. Discrete Contin. Dynam. Systems 8, 289-302 (2002)
3. N. Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94, 1-7 (2000)
4. BIRINDELLI, I., MITIDIERI, E.: Liouville theorems for elliptic inequalities and applications. Proc. Roy. Soc. Edinburgh 128A, 1217-1247 (1998)
5. t - Δu = g(u) revisited. Adv. Differ. Equations 1, 73-90 (1996)
6. BREZIS, H., TURNER, R.: On a class of superlinear elliptic problems. Commun. Partial Diff. Equations 2, 601-614 (1977)
7. BUSCA, J., MANASEVICH, R.: A Liouville-type theorem for Lane-Emden system. Indiana Univ. Math. J. 51, 37-51 (2002)
8. CHEN, H.: Positive steady-state solutions of a non-linear reaction-diffusion system. Math. Meth. Appl. Sci. 20, 625-634 (1997)
9. CLEMENT, PH., DE FIGUEIREDO, D.G., MITIDIERI, E.: Positive solutions of semilinear elliptic systems. Commun. Partial Diff. Equations 17, 923-940 (1992)
10. CLEMENT, PH., DE FIGUEIREDO, D.G., MITIDIERI, E.: A priori estimates for positive solutions of semilinear elliptic systems via Hardy-Sobolev inequalities. Pitman Res. Notes Math. 343, 73-91, 1996
11. CLEMENT, PH., FLECKINGER, J., MITIDIERI, E., DE THÉLIN, F.: Existence of positive solutions for nonvariational quasilinear system. J. Diff. Equations 166, 455-477 (2000)
12. CLEMENT, PH., MANASEVICH, R., MITIDIERI, E.: Positive solutions for a quasilinear system via blow up. Comm. Partial Diff. Equations 18, 2071-2106 (1993)
13. COSNER, C.: Positive solutions for a superlinear elliptic systems without variational structure. Nonlinear Anal. 8, 1427-1436 (1984)
14. CUESTA, M., DE FIGUEIREDO, D.G., SRIKANT, P.N.: On a resonant-superlinear elliptic problem. Calc. Var. PDE 17, 221-233 (2003)
15. DALL'ACQUA, A.: Positive solutions for a class of reaction-diffusion systems. Commun. Pure Appl. Anal. 2, 65-76 (2003)
16. DICKSTEIN, F., ESCOBEDO, M.: A maximum principle for semilinear parabolic systems and applications. Nonlinear Anal. 45, 825-837 (2001)
17. DE FIGUEIREDO, D.G.: Semilinear elliptic systems. Nonlinear Functional Analysis and Applications to Differential Equations, Trieste 1997, World Sci. Publishing, River Edge, N.J., 1998, pp. 122-152
18. DE FIGUEIREDO, D.G., FELMER, P.: A Liouville-type theorem for elliptic systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21, 387-397 (1994)
19. DE FIGUEIREDO, D.G., LIONS, P.-L., NUSSBAUM, R.D.: A priori estimates and existence of positive solutions of semilinear elliptic equations. J. Math. Pures et Appl. 61, 41-63 (1982)
20. DE FIGUEIREDO, D.G., YANG, J.: A priori bounds for positive solutions of a non-variational elliptic system. Commun. Partial Diff. Equations 26, 2305-2321 (2001)
21. q spaces and universal bounds for global solutions. Math. Annalen 320, 87-113 (2001)
22. GIDAS, B., SPRUCK, J.: A priori bounds for positive solutions of a nonlinear elliptic equations. Commun. Partial Diff. Equations 6, 883-901 (1981)
23. GU, Y., WANG, M.: A semilinear parabolic system arising in the nuclear reactors. Chinese Sci. Bull. 39, 1588-1592 (1994)
24. Gu, Y., WANG, M.: Existence of positive stationary solutions and threshold results for a reaction-diffusion system. J. Diff. Equations 130, 277-291 (1996)
25. LI, Y., LIU, Q., XIE, C.: Semilinear reaction-diffusion systems of several components. J. Diff. Equations 187, 510-519 (2003)
26. LOU, Y.: Necessary and sufficient condition for the existence of positive solutions of certain cooperative system. Nonlinear Anal. 26, 1079-1095 (1996)
27. MITIDIERI, E.: Nonexistence of positive solutions of semilinear elliptic systems in RN. Diff. Integral Equations 9, 465-479 (1996)
28. QUITTNER, P.: Transition from decay to blow-up in a parabolic system. Archivum Math. 34, 199-206 (1998)
29. REICHEL, W., ZOU, H.: Non-existence results for semilinear cooperative elliptic systems via moving spheres. J. Diff. Equations 161, 219-243 (2000)
30. SERRIN, J., ZOU, H.: Non-existence of positive solutions of Lane-Emden systems. Diff. Integral Equations 9, 635-653 (1996)
31. SERRIN, J., ZOU, H.: Existence of positive solutions of the Lane-Emden system. Atti Sem. Mat. Fis. Univ. Modena 46, suppl., 369-380 (1998)
32. p spaces. To appear
33. SOUTO, M.: A priori estimate and existence of positive solutions of nonlinear cooperative elliptic system. Diff. Integral Equations 8, 1245-1258 (1995)
34. DE THÉLIN, F., VÉLIN, J.: Existence and nonexistence of nontrivial solutions for some nonlinear elliptic systems. Rev. Mat. Univ. Complut. Madrid 6, 153-194 (1993)
35. TROY, W.C.: Symmetry properties in systems of semilinear elliptic equations. J. Diff. Equations 42, 400-413 (1981)
36. ZHENG, S.: Nonexistence of positive solutions to a semilinear elliptic system and blow-up estimates for a reaction-diffusion system. J. Math. Anal. Appl. 232, 293-311 (1999)
37. ZOU, H.: A priori estimates for a semilinear elliptic systems without variational structure and their applications. Math. Ann. 323, 713-735 (2002)